\(\sum _{ij} A_{ij} = 2E\)  is the total number of edges, \(k_i\)  is the degree of the node \(i\)  and \(\gamma \)  is the so-called resolution parameter [1]}, set to 1 throughout this work. The sum over \(A_{ij}\)  measures the fraction of within community edges in the observed network and the sum over degrees gives the fraction of within community edges expected under a degree preserving randomization of the network, known as the configuration model [2]}. Despite some well-known issues identifying small communities (the so-called resolution limit [3]}) modularity maximization is the basis for a number of very popular community detection algorithms e.g. [4]}, [5]}.
